Minimal Binary Abelian Codes of length $p^m q^n$

TL;DR

This paper investigates minimal binary abelian codes of length $p^m q^n$, identifying their generating idempotents and weight bounds, with examples demonstrating the bounds' attainability.

Contribution

It provides a detailed characterization of minimal binary abelian codes of length $p^m q^n$, including explicit idempotents and weight bounds under certain conditions.

Findings

Identified idempotents generating minimal codes

Established bounds for code weights

Provided examples where bounds are attained

Abstract

We consider binary abelian codes of length $p^m q^n$, where $p$ and $q$ are prime rational integers under some restrictive hypotheses. In this case, we determine the idempotents generating minimal codes and either the respective weights or bounds of these weights. We give examples showing that these bounds are attained in some cases.